If a cube number measure a cube number, the side will also measure the side; and, if the side measure the side, the cube will also measure the cube. For let the cube number A measure the cube B, and let C be the side of A and D of B; I say that C measures D. For let C by multiplying itself make E, and let D by multiplying itself make G; further, let C by multiplying D make F, and let C, D by multiplying F make H, K respectively. Now it is manifest that E, F, G and A, H, K, B are continuously proportional in the ratio of C to D. [VIII. 11, 12] And, since A, H, K, B are continuously proportional, and A measures B, therefore it also measures H. [VIII. 7] And, as A is to H, so is C to D; therefore C also measures D. [VII. Def. 20] Next, let C measure D; I say that A will also measure B. For, with the same construction, we can prove in a similar manner that A, H, K, B are continuously proportional in the ratio of C to D.